Chaotic Dynamical Ferromagnetic Phase Induced by Nonequilibrium Quantum Fluctuations

Alessio Lerose, Jamir Marino, Bojan Žunkovič, Andrea Gambassi, and Alessandro Silva

Phys. Rev. Lett. 120, 130603 – Published 30 March 2018

Abstract

We investigate the robustness of a dynamical phase transition against quantum fluctuations by studying the impact of a ferromagnetic nearest-neighbor spin interaction in one spatial dimension on the nonequilibrium dynamical phase diagram of the fully connected quantum Ising model. In particular, we focus on the transient dynamics after a quantum quench and study the prethermal state via a combination of analytic time-dependent spin wave theory and numerical methods based on matrix product states. We find that, upon increasing the strength of the quantum fluctuations, the dynamical critical point fans out into a chaotic dynamical phase within which the asymptotic ordering is characterized by strong sensitivity to the parameters and initial conditions. We argue that such a phenomenon is general, as it arises from the impact of quantum fluctuations on the mean-field out of equilibrium dynamics of any system which exhibits a broken discrete symmetry.

Received 15 June 2017
Revised 22 December 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.130603

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics Quantum phase transitions Spin waves

Quantum spin models

Statistical Physics & Thermodynamics

Authors & Affiliations

Alessio Lerose^1,2, Jamir Marino³, Bojan Žunkovič⁴, Andrea Gambassi^1,2, and Alessandro Silva¹

¹SISSA—International School for Advanced Studies, via Bonomea 265, I-34136 Trieste, Italy
²INFN—Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34136 Trieste, Italy
³Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany
⁴Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

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Vol. 120, Iss. 13 — 30 March 2018

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Images

Figure 1
A dynamical phase diagram of the model in Eq. (1) after a quantum quench starting from the ferromagnetic ground state with $g = 0$ and positive expectation value $S_{x} > 0$ of the global magnetization, in the plane of the postquench value $g$ of the transverse field and $J$ of the nearest-neighbor coupling (here $N = 100$ ). We consider here the range of values of $g$ and $J$ within which the low-density spin wave expansion is applicable, and units are chosen such that $\bar{λ} \equiv λ + J = 1$ . The color of each point of the diagram is determined by the value of long-time average ${\bar{S}}_{x}$ of $S_{x}$ : light yellow for ${\bar{S}}_{x} > 0$ , orange for ${\bar{S}}_{x} = 0$ , and blue for ${\bar{S}}_{x} < 0$ . Regions A and B correspond to the dynamic ferromagnetic and paramagnetic phase, respectively, of the mean-field model ( $J = 0$ ). Upon increasing $J$ at a fixed $g$ close to the mean-field critical point, i.e., $g ≃ \bar{λ}$ , a new chaotic dynamical ferromagnetic phase C arises, exhibiting relaxation from an initial paramagnetic behavior to symmetry-broken sectors [process (a) in the inset] sometimes followed by assisted hopping between the two sectors with opposite signs of ${\bar{S}}_{x}$ [process (b) in the inset]. See Fig. 2 for an illustration of the dynamics in region C.
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Figure 2
The evolution of the order parameter $S_{x} (t)$ in the chaotic dynamical ferromagnetic phase (indicated by C in Fig. 1) for $\bar{λ} \equiv λ + J = 1$ , $g = 1.03$ , with $J = 0.1$ (solid red) and $J = 0.1001$ (dashed blue), i.e., two very close points in the nonequilibrium phase diagram, located at the ending point of the black arrow in Fig. 1 (here $N = 200$ ). The dynamical order parameter $S_{x} (t)$ initially displays a paramagnetic behavior, with a gradual loss of energy in favor of the creation of spin waves, witnessed by a growth of $ε (t)$ . This makes the orbit fall into one of the two ferromagnetic wells, corresponding to process (a) of Fig. 1. However, it might later reabsorb some spin waves and hop to the opposite ferromagnetic sector, corresponding to process (b) of Fig. 1. The two lines are practically on top of each other during the initial paramagnetic transient, but show completely different fates at the onset of the critical process (a) and they eventually end up in distinct wells. [In both cases $ε (t)$ grows from $ε (t = 0) = 0$ to values around 0.04 in the final stage.] Such extreme sensitivity illustrates the “mosaic” appearance of the region C in Fig. 1.
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Figure 3
The evolution of $S_{x}$ for $J = 0.67$ , $g = 0.5$ , 0.83, 1, 1.33 (red, green, blue, gray), with $\bar{λ} = 1$ and $N = 400$ , as obtained from MPS-TDVP simulations. Inset: The sensitivity of $S_{x} (t)$ to the system size $N$ in the chaotic dynamical ferromagnetic phase, for a system with $J = 0.5$ , $\bar{λ} = 1$ , $g = 1.1$ , and a bond dimension $D = 128$ . $S_{x}$ approaches a positive value for small $N = 123$ , 124 (dashed, dotted). However, upon adding just one spin ( $N = 125$ , dash-dotted), $S_{x}$ reverses its sign, and ${\bar{S}}_{x}$ converges to a negative value, which is also observed in larger systems with $N = 400$ (solid line). For further details see [25].
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